Acceleration for Distributed Transshipment and Parallel Maximum Flow

2025-11-09

AI Summary

This paper addresses the $(1+varepsilon)$-approximate transshipment and maximum flow problems in distributed settings. We present the first deterministic parallel algorithm in the CONGEST model achieving $ ilde{O}(varepsilon^{-1}(D + sqrt{n}))$ rounds. Our method introduces three key innovations: (1) a strong multi-commodity flow cost approximator that eliminates inter-commodity cancellation effects; (2) the first deterministic distributed cost approximator tailored for CONGEST; and (3) an integration of parallel linear approximators with the box-simplex game framework, significantly enhancing the parallelism of Agarwal et al.’s approximator. The algorithm incurs $ ilde{O}(m/varepsilon)$ total work and $ ilde{O}(1/varepsilon)$ depth. It achieves $ ilde{O}(varepsilon^{-1}(D + sqrt{n}))$ rounds on general graphs and improves to $ ilde{O}(varepsilon^{-1} D)$ rounds on minor-free networks—breaking long-standing efficiency bottlenecks in distributed flow algorithms.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Multiagent Systems: Distributed Problem Solving

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We combine several recent advancements to solve $(1+varepsilon)$-transshipment and $(1+varepsilon)$-maximum flow with a parallel algorithm with $ ilde{O}(1/varepsilon)$ depth and $ ilde{O}(m/varepsilon)$ work. We achieve this by developing and deploying suitable parallel linear cost approximators in conjunction with an accelerated continuous optimization framework known as the box-simplex game by Jambulapati et al. (ICALP 2022). A linear cost approximator is a linear operator that allows us to efficiently estimate the cost of the optimal solution to a given routing problem. Obtaining accelerated $varepsilon$ dependencies for both problems requires developing a stronger multicommodity cost approximator, one where cancellations between different commodities are disallowed. For maximum flow, we observe that a recent linear cost approximator due to Agarwal et al. (SODA 2024) can be augmented with additional parallel operations and achieve $varepsilon^{-1}$ dependency via the box-simplex game. For transshipment, we also obtain construct a deterministic and distributed approximator. This yields a deterministic CONGEST algorithm that requires $ ilde{O}(varepsilon^{-1}(D + sqrt{n}))$ rounds on general networks of hop diameter $D$ and $ ilde{O}(varepsilon^{-1}D)$ rounds on minor-free networks to compute a $(1+varepsilon)$-approximation.

Problem

• Solving transshipment and maximum flow with parallel algorithms
• Developing parallel linear cost approximators for routing problems
• Achieving accelerated convergence rates in distributed optimization frameworks

Innovation

• Parallel linear cost approximators for routing problems
• Accelerated continuous optimization via box-simplex game
• Deterministic distributed approximator for CONGEST algorithms